The Bluebirds first made allegations in August 2014 and are considering taking the case further if the FA does not report soon.
The FA said: "Because of the emergence of significant new information the FA is currently not in a position to conclude its ongoing enquiries into... Malky Mackay and Iain Moody."
Moody was also caught up in the offensive texts allegations following his time as head of recruitment at Cardiff under Mackay, who is now Wigan manager.
Mackay has admitted sending offensive messages and apologised,adding he is not racist, sexist, homophobic or anti-Semitic.
The options Cardiff are considering are understood to include:
- Reporting the matter to world football's governing body Fifa.
- Going to the European Court of Arbitration for Sport.
- Seeking a judicial review over the FA rules.
- Going public with the evidence they have submitted to the FA.
Mackay was sacked by Cardiff in December 2013 and is now manager of Championship rivals Wigan Athletic.
Owner Vincent Tan had ordered members of the club board not to attend Cardiff's game at Wigan on 24 February in an official capacity. Cardiff won 1-0.